Spin(7) holonomy manifold and superconnection (Q2716831)

An eight-dimensional generalization of gravitational instantons is formulated by using volume-preserving vector fields. Geometrically, gravitational instantons are intimately linked to Riemannian manifolds with special holonomy groups. The authors provide a new construction of Spin(7) holonomy manifolds including a brief review of the geometry of these manifolds. An explicit example of Ricci-flat metric with holonomy in Spin(7) on the space \(\mathbb{R}^2\times S^3\times S^3\) is presented. This is the same metric constructed by other authors by using different methods [see \textit{R. L. Bryant} and \textit{S. M. Salamon}, Duke Math. J. 58, 829-850 (1989; Zbl 0681.53021); \textit{G. W. Gibbons, D. N. Page} and \textit{C. N. Pope}, Commun. Math. Phys. 127, 529-553 (1990; Zbl 0699.53053)]. Using the concept of superconnection introduced by \textit{D. Quillen} [see Topology 24, 89-95 (1985; Zbl 0569.58030)], it is shown that their formulation has a natural interpretation in the Chern-Simons theory. Finally, \(G_2\) holonomy manifolds (seven-dimensional gravitational instantons) are briefly discussed.